Mr. Stone, a veteran political consultant, swapped private Twitter messages last year with “Guccifer 2.0,” the website reported Wednesday citing an unnamed source. If accurate, the allegation means Mr. Trump’s longtime confidante and one-time presidential campaign manager spoke in private with a persona intimately implicated in the operation that helped dash Democratic nominee Hillary Clinton’s chance at the White House, and would undeniably amplify existing concerns regarding Team Trump’s alleged ties to Russia.

Guccifer 2.0 first emerged in June 2016 a day after The Washington Post reported that hackers likely acting on behalf of the Russian government had infiltrated the DNC’s computer network. In what began with a blog post, the persona — a self-described Romanian hacktivist — insisted that they alone had breached the DNC, and had taken a treasure trove of data subsequently supplied to WikiLeaks.

Indeed, the secret-spilling website began publishing internal Democratic Party correspondence the following month, including embarrassing documents pounced upon by Mrs. Clinton’s presidential opponent.

In addition to a handful of public, documented Twitter exchanges between Mr. Stone and the Guccifer 2.0 person, The Smoking Gun indicated Wednesday that federal investigators have obtained private, “direct messages” involving both accounts. The nature of the alleged correspondence was not made clear, and Mr. Stone didn’t deny they may have taken place: When asked if the conversations occurred, Mr. Stone wrote “don’t recall” in a text to The Smoking Gun.

“Numerous people who work for me have access to my twitter feed,” he added.

Guccifer 2.0’s Twitter account was in regular contact with multiple reporters prior to last year’s election, including those employed by The Smoking Gun. According to Wednesday’s report, the website grilled Guccifer 2.0 last August concerning any possible conversations he had with Mr. Stone in private, the likes of which have reportedly since been obtained by federal investigators.

The FBI began investigating Guccifer 2.0 last year and obtained detailed records from Twitter that would include tweets and direct messages involving the persona, The Smoking Gun reported, citing two sources. The FBI is separately in the midst of investigating potential ties between individuals close to Mr. Trump, including Mr. Stone, and Moscow.

U.S. intelligence officials concluded in January that Russian intelligence deployed the Guccifer 2.0 hacker to distribute documents stolen in cyberattacks waged against the DNC, and said it appeared that more than one person had used the persona.

The White House and Kremlin have previously denied involvement in the DNC hack.